Triviality conjecture for diagonal smooth projectors

Let $n\in\mathbb{N}^*$, and let $\{\mathfrak{b}_{m,n}\}_{m\geq0}$ be a sequence satisfying

$$\mathfrak{b}_{m,n}=\sum_{r=0}^m\sum_{h=0}^r\left(\frac{m!}{(m-r)!(r-h)!h!}\right)^n\mathfrak{b}_{m+h-r,n}\mathfrak{b}_{r,n}=\sum_{h=0}^\infty(-1)^{(m+h)n}\binom{m+h}{m}^n\overline{\mathfrak{b}}_{m+h,n},\qquad \forall m\in\mathbb{N}.$$

These sequences encode the diagonal elements of $P_0(A^\infty(\mathbb{R}^{2n}_\Theta))$. The diagonal projector triviality conjecture.

$$\mathfrak{b}_{m,n}=\begin{cases}0\text{ or }1,&m=0,\\0,&m\geq1,\end{cases}$$

which would imply $P_0(A^\infty(\mathbb{R}^{2n}_\Theta))=\{0,1\}$ for every $n\in\mathbb{N}^*$. The source describes this as a conjecture motivated by computations and leaves it open.